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A square is formed by joining four rods each of mass M and length l .it's moment of inertia about an axis pq in its plane and passing through one of its corners
Asked by francojoem | 27 Dec, 2019, 04:05: PM Expert Answer Figure shows four identical rods joined together in the form of square and is rotated about an axis PQ passing through a corner A.

The rods are of equal length l and have equal mass M.

Let us calculate moment of inertia of rod AB. Consider a small element ds at a distance s from corner A.
If ρ is linear mass density , i.e., mass per unit length then moment of inertia IAB of rod AB is given by ............................(1)
Similarly moment of inertia IBC of rod BC is given by .......................(2)

above integration is performed using substitution,  u = [ (l/√2) + (s/√2) ]  , du = (1/√2)ds

lower limit = l/√2  and upper limit = √2l

we get, IBC = (7/6) Ml2

Moment of inertia of rod AD is same as that of rod AB . Similarly , moment of inertia of rod DC is same as that of rod BC.

Hence, total moment of inertia I of system is given by,

I = IAB + IBC + IAD + IDC = (1/6)Ml2 + (7/6)Ml2 + (1/6)Ml2 + (7/6)Ml2 = (8/3)Ml2
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